Chapter 11

Stratification

SUMMARY : After having studied the effects of rotation in homogeneous fluids, we nowturn our attention toward the other distinctive feature of geophysical fluid dynamics, namely,stratification. A basic measure of stratification, the Brunt–Vaisala frequency, is introduced,and the accompanying dimensionless ratio, the Froude number, is defined and given a phys-ical interpretation. The numerical part deals with the handling of unstable stratification inmodel simulations.

11.1 Introduction

As Chapter1 stated, problems in geophysical fluid dynamics concern fluidmotions withone or both of two attributes, namely, ambient rotation and stratification. In the precedingchapters, attention was devoted exclusively to the effectsof rotation, and stratification wasavoided by the systematic assumption of a homogeneous fluid.We noted that rotation impartsto the fluid a strong tendency to behave in a columnar fashion —to be vertically rigid.

By contrast, a stratified fluid, consisting of fluid parcels ofvarious densities, will tendunder gravity to arrange itself so that the higher densitiesare found below lower densities.This vertical layering introduces an obvious gradient of properties in the vertical direction,which affects — among other things — the velocity field. Hence, the vertical rigidity inducedby the effects of rotation will be attenuated by the presenceof stratification. In return, thetendency of denser fluid to lie below lighter fluid imparts a horizontal rigidity to the system.

Because stratification induces a certain degree of decoupling between the various fluidmasses (those of different densities), stratified systems typically contain more degrees offreedom than homogeneous systems, and we anticipate that the presence of stratificationpermits the existence of additional types of motions. When the stratification is mostly ver-tical (e.g., layers of various densities stacked on top of one another),gravity waves can besustained internally (Chapter13). When the stratification also has a horizontal component,additional waves can be permitted. These may lead to motion in equilibrium (Chapter15), or,if they grow at the expense of the basic potential energy available in the system, may cause

319

320 CHAPTER 11. STRATIFICATION

instabilities (Chapter17).

11.2 Static stability

Let us first consider a fluid in static equilibrium. Lack of motion can occur only in the absenceof horizontal forces and thus in the presence of horizontal homogeneity. Stratification is thenpurely vertical (Figure11-1).

h

ρ(z + h)

z ρ(z)

Figure 11-1 When an incompressiblefluid parcel of densityρ(z) is verticallydisplaced from levelz to level z + hin a stratified environment, a buoyancyforce appears because of the densitydifferenceρ(z)−ρ(z+h) between theparticle and the ambient fluid.

It is intuitively obvious that if the heavier fluid parcels are found below the lighter fluidparcels, the fluid is stable, whereas if heavier parcels lie above lighter ones, the system is aptto overturn, and the fluid is unstable. Let us now verify this intuition. Take a fluid parcel at aheightz above a certain reference level, where the density isρ(z), and displace it verticallyto the higher levelz + h, where the ambient density isρ(z + h) (Figure11-1). If the fluidis incompressible, our displaced parcel retains its formerdensity despite a slight pressurechange, and at that new level is subject to a net downward force equal to its own weightminus, by Archimedes’ buoyancy principle, the weight of thedisplaced fluid, thus

g [ρ(z) − ρ(z + h)] V,

whereV is the volume of the parcel. As it is written, this force is positive if it is directeddownward. Newton’s law (mass times acceleration equals upward force) yields

ρ(z) Vd2h

dt2= g [ρ(z + h) − ρ(z)] V. (11.1)

Now, geophysical fluids are generally only weakly stratified; the density variations, al-though sufficient to drive or affect motions, are nonetheless relatively small compared to theaverage or reference density of the fluid. This remark was theessence of the Boussinesqapproximation (Section3.7). In the present case, this fact allows us to replaceρ(z) on theleft-hand side of (11.1) by the reference densityρ0 and to use a Taylor expansion to approxi-mate the density difference on the right by

ρ(z + h) − ρ(z) 'dρ

dzh.

After a division byV , equation (11.1) reduces to

11.3. ATMOSPHERIC STRATIFICATION 321

d2h

dt2−

g

ρ0

dρ

dzh = 0, (11.2)

which shows that two cases can arise. The coefficient−(g/ρ0)dρ/dz is either positive ornegative. If it is positive (dρ/dz < 0, corresponding to a fluid with the greater densitiesbelow the lesser densities), we can define the quantityN2 as

N2 = −g

ρ0

dρ

dz, (11.3)

and the solution to the equation has an oscillatory character, with frequencyN . Physically,this means that, when displaced upward, the parcel is heavier than its surroundings, feels adownward recalling force, falls down, and, in the process, acquires a vertical velocity; uponreaching its original level the particle’s inertia causes it to go further downward and to becomesurrounded by heavier fluid. The parcel, now buoyant, is recalled upward, and oscillationspersist about the equilibrium level. The quantityN , defined by the square root of (11.3),provides the frequency of the oscillation and can thus be called thestratification frequency.It goes more commonly, however, by the name of Brunt–Vaisala frequency, in recognition ofthe two scientists who were the first to highlight the importance of this frequency in stratifiedfluids. (See their biographies at the end of this chapter.)

If the coefficient in equation (11.1) is negative (i.e., dρ/dz > 0, corresponding to a top-heavy fluid configuration), the solution exhibits exponential growth, a sure sign of instability.The parcel displaced upward is surrounded by heavier fluid, finds itself buoyant, and movesfarther and farther away from its initial position. Obviously, small perturbations will ensurenot only that the single displaced parcel will depart from its initial position, but that all otherfluid parcels will likewise participate in a general overturning of the fluid until it is finallystabilized, with the lighter fluid lying above the heavier fluid. If, however, a permanentdestabilization is forced onto the fluid, such as by heating from below or cooling from above,the fluid will remain in constant agitation, a process calledconvection.

11.3 A note on atmospheric stratification

In a compressible fluid, such as the air of our planetary atmosphere, density can change inone of two ways: by pressure changes or by internal-energy changes. In the first case, apressure variation resulting in no heat exchange (i.e., an adiabatic compression or expansion)is accompanied by both density and temperature variations:All three quantities increase (ordecrease) simultaneously, though not in equal proportions. If the fluid is made of fluid parcelsall having the same heat content, the lower parcels, feelingthe weight of those above them,will be more compressed than those in the upper levels, and the system will appear stratified,with the denser and warmer fluid underlying the lighter, colder fluid. But such stratificationcannot be dynamically relevant, for if parcels are interchanged adiabatically, they adjust theirdensity and temperature according to the local pressure, and the system is left unchanged.

In contrast, internal-energy changes are dynamically important. In the atmosphere, suchvariations occur because of a heat flux (such as heating in thetropics and cooling at high

322 CHAPTER 11. STRATIFICATION

latitudes, or according to the diurnal cycle) or because of variations in air composition (suchas water vapor). Such variations among fluid parcels do remain despite adiabatic compres-sion/expansion and cause density differences that drive motions. It is thus imperative todistinguish, in a compressible fluid, the density variations that are dynamically relevant fromthose that are not. Such separation of density variations leads to the concept of potentialdensity.

First, we consider a neutral (adiabatic) atmosphere – that is, one consisting of all airparcels having the same internal energy. Further, let us assume that the air, a mixture ofvarious gases, behaves as a single ideal gas. Under these assumptions, we can write theequation of state and the adiabatic conservation law:

p = RρT, (11.4)

p

p0=

(

ρ

ρ0

)γ

, (11.5)

wherep, ρ andT are, respectively, the pressure, density1 and absolute temperature;R =Cp − Cv andγ = Cp/Cv are the constants of an ideal gas2. Finally,p0 andρ0 are referencepressure and density characterizing the level of internal energy of the fluid; the correspondingreference temperatureT0 is obtained from (11.4) — that is,T0 = p0/Rρ0. Expressing bothpressure and density in terms of the temperature, we obtain

p

p0=

(

T

T0

)γ/(γ−1)

(11.6a)

ρ

ρ0=

(

T

T0

)1/(γ−1)

. (11.6b)

Without motion, the atmosphere is in static equilibrium, which requires hydrostatic bal-ance:

dp

dz= − ρg. (11.7)

Elimination ofp andρ by use of (11.6a) and (11.6b) yields a single equation for the temper-ature:

dT

dz= −

γ − 1

γ

g

R

= −g

Cp. (11.8)

In the derivation, it was assumed thatp0, ρ0, and thusT0 are not dependent onz, in agreementwith our premise that the atmosphere is composed of parcels with identical internal-energycontents. Equation (11.8) states that the temperature in such atmosphere must decrease with

1In contrast with preceding chapters, the variablesp andρ denote here the full pressure and density.2For air, values are:Cp = 1005 J kg−1 K−1, Cv = 718 J kg−1 K−1, R = 287 J kg−1 K−1, andγ = 1.40.

11.3. ATMOSPHERIC STRATIFICATION 323

increasing height at the uniform rateg/Cp ' 10 K/km. This gradient is called theadiabaticlapse rate. Physically, lower parcels are under greater pressure thanhigher parcels and thushave higher densities and temperatures. This explains why the air temperature is lower onmountain tops than in the valleys below.

It almost goes without saying that the departures from this adiabatic lapse rate — andnot the actual temperature gradients — are to be considered in the study of atmosphericmotions. We can demonstrate this clearly by redoing here, with a compressible fluid, theanalysis of a vertical displacement performed in the previous section with an incompressiblefluid. Consider a vertically stratified gas with pressure, density, and temperature,p, ρ andT , varying with heightz but not necessarily according to (11.8); that is, the heat content inthe fluid is not uniform. The fluid is in static equilibrium so that equation (11.7) is satisfied.Consider now a parcel at heightz; its properties arep(z), ρ(z) andT (z). Imagine then thatthis fluid parcel is displaced adiabatically upward over a small distanceh. According to thehydrostatic equation, this results in a pressure changeδp = −ρgh, which causes density andtemperature changes given by the adiabatic constraints (11.5) and (11.6a): δρ = −ρgh/γRTandδT = −(γ − 1)gh/γR. Thus, the new density isρ′ = ρ+ δρ = ρ− ρgh/γRT . But, atthat new level, the ambient density is given by the stratification: ρ(z+h) ' ρ(z)+(dρ/dz)h.The net force exerted on the parcel is the difference betweenits own weight and the weightof the displaced fluid at the new location (the buoyancy force), which per volume is

F = g [ρambient − ρparcel]

= g [ρ(z + h) − ρ′]

' g

(

dρ

dz+

ρg

γRT

)

h.

As the ideal-gas law (p = RρT ) holds everywhere, the vertical gradients of pressure, densityand temperature are related by:

dp

dz= RT

dρ

dz+ Rρ

dT

dz.

With the pressure gradient given by the hydrostatic balance(11.7), it follows that the densityand temperature gradients are related by

1

ρ

dρ

dz+

1

T

dT

dz+

g

RT= 0,

and the force on the fluid parcel can be expressed in terms of the temperature gradient

F ' −ρg

T

(

dT

dz+

g

Cp

)

h.

If

N2 = −g

ρ

(

dρ

dz+

ρg

γRT

)

(11.9a)

= +g

T

(

dT

dz+

g

Cp

)

(11.9b)

324 CHAPTER 11. STRATIFICATION

is a positive quantity, the force recalls the particle towards its initial level, and the stratificationis stable. As we can clearly see, the relevant quantity is notthe actual temperature gradient butits departure from the adiabatic gradientg/Cp. As in the previous case of a stably stratifiedincompressible fluid, the quantityN is the frequency of vertical oscillations. It is called thestratification, or Brunt–Vaisala, frequency.

In order to avoid the systematic subtraction of the adiabatic gradient from the tempera-ture gradient, the concept of potential temperature is introduced. Thepotential temperature,denoted byθ, is defined as the temperature that the parcel would have if itwere broughtadiabatically to a given reference pressure3. From (11.6a), we have

p

p0=

(

T

θ

)γ/(γ−1)

and hence

θ = T

(

p

p0

)

−(γ−1)/γ

. (11.10)

The corresponding density is called thepotential density, denotedσ:

σ = ρ

(

p

p0

)

−1/γ

. (11.11)

The definition of the stratification frequency (11.9b) takes the more compact form

N2 = −g

σ

dσ

dz= +

g

θ

dθ

dz. (11.12)

Comparison with the earlier definition, (11.3), immediately shows that the substitution ofpotential density for density allows us to treat compressible fluids as incompressible.

During daytime and above land, the lower atmosphere is typically heated from below bythe warmer ground and is in a state of turbulent convection. The convective layer not onlycovers the region where the time-averaged gradient of potential temperature is negative butalso penetrates into the region above where it is positive (Figure 11-2). Consequently, thesign ofN2 at a particular level is not unequivocally indicative of stability at that level. Forthis reason, Stull (1991) advocates the use of a non-local criterion to determine static stability.Those considerations apply equally well to the upper ocean under surface cooling.

When the air is moist, the thermodynamics of water vapor affect the situation, and, be-cause the value ofCp for water vapor is higher than that for dry air, the adiabaticlapse rate isreduced. As the temperature of ascending air drops, the relative humidity may reach 100%,in which case condensation occurs and water droplets form a cloud. Condensation liberateslatent heat, which reduces the temperature drop if parcels continue to ascend. The lapse isthen further reduced to asaturated adiabatic lapse rate, as depicted in Figure11-3.

3In the atmosphere, this reference pressure is usually takenas the standard sea-level pressure of 1013.25 millibars= 1.01325× 105 N/m2.

11.3. ATMOSPHERIC STRATIFICATION 325

Figure 11-2 Typical profile of potential temperature in the lower atmosphere above warm ground.Heating from below destabilizes the air, generating convection and turbulence. Note how the convectivelayer extends not only over the region of negativeN2 but also slightly beyond, whereN2 is positive.Such a situation shows that a positive value ofN2 may not always be indicative of local stability. Globalstability refers then to regions where even a finite amplitude displacement cannot destabilize the fluidparcel. (From Stull, 1991)

326 CHAPTER 11. STRATIFICATION

T

z

Adiabatic lapse rate

Condensation level

Upper cloud level

Saturated lapse rate

Air temperature

k

Figure 11-3 Fluid parcels locatedaround level z amidst a tempera-ture gradient (curved solid line) lo-cally exceeding the adiabatic lapse rate(dashed line) are in an unstable situa-tion. They move upward and eventu-ally reach their saturation level, con-densation takes place, and the lapse rateis decreased. If an inversion is presentat higher levels, cloud extension is ver-tically limited.

11.4 Convective adjustment

When gravitational instability is present in the ocean or atmosphere, non-hydrostatic move-ments tend to restore stability through narrow columns of convection, rising plumes and ther-mals in the atmosphere and so-called convective chimneys inthe ocean (e.g., Marshall andSchott, 1999). These vigorous vertical motions are not resolved by most computer models,and parameterizations calledconvection schemesare introduced to remove the instability andmodel the mixing associated with convection. Such parameterization can be achieved by ad-ditional terms in the governing equations, typically through a much increased eddy viscosityand diffusivity wheneverN2 ≤ 0 (e.g., Cox, 1984; Marotzke, 1991). Other parameterizationsare pieces of computer code of the type (see Figure11-4):

while there is any denser fluid being on top of lighter fluidloop over all layers

if density of layer above > density of layer belowmix properties of both layers, with a volume-weighted average

end ifend loop over all layers

end while

Oceanic circulation models (e.g., Bryan, 1969; Cox, 1984) were the first to use this type ofparameterization.

The mixing accomplished by such scheme, however, is too strong in practice, becausethe model mixes fluid properties instantaneously over an entire horizontal grid cell of size

11.5. FROUDE NUMBER 327

Figure 11-4 Illustration of convec-tive adjustment within a fluid heatedfrom below. Grid boxes below heav-ier neighbors are systematically mixedin pairs until the whole fluid column isrendered stable.

∆x∆y, while physical convection operates at shorter scales and only partially mixes thephysical properties at the spot. Therefore, numerical mixing should preferably be replacedby a mere swapping of fluid masses, under the assumption that convection carries part ofthe properties without alteration to their new level of equilibrium (e.g., Roussenovet al.,2001). It is clear that some arbitrariness remains and that every application demands its owncalibration. Among other things, changing the time step clearly modifies the speed at whichmixing takes place.

In atmospheric applications, the situation is more complicated as it may involve con-densation, latent-heat release, and precipitation duringconvective movement. Atmosphericconvection parameterizations involve delicate adjustments of both temperature and moisturein the vertical (e.g., Kuo, 1974; Betts, 1986).

11.5 The importance of stratification: The Froude number

It was established in Section1.5 that rotational effects are dynamically important when theRossby number is on the order of unity or less. This number compares the distance traveledhorizontally by a fluid parcel during one revolution (∼ U/Ω) with the length scale overwhich the motions take place (L). Rotational effects are important when the former is lessthan the latter. By analogy, we may ask whether there exists asimilar number measuring theimportance of stratification. From the remarks in the preceding sections, we can anticipatethat the stratification frequency,N , and the height scale,H , of a stratified fluid will play rolessimilar to those ofΩ andL in rotating fluids.

To illustrate how such a dimensionless number can be derived, let us consider a stratifiedfluid of thicknessH and stratification frequencyN flowing horizontally at a speedU overan obstacle of lengthL and height∆z (Figure11-5). We can think of a wind in the loweratmosphere blowing over a mountain range. The presence of the obstacle forces some ofthe fluid to be displaced vertically and, hence, requires some supply of gravitational energy.Stratification will act to restrict or minimize such vertical displacements in some way, forcingthe flow to pass around rather than over the obstacle. The greater the restriction, the greaterthe importance of stratification.

The time passed in the vicinity of the obstacle is approximately the time spent by a fluid

328 CHAPTER 11. STRATIFICATION

Figure 11-5 Situation in which a stratified flow encounters an obstacle, forcing some fluid parcels tomove vertically against a buoyancy force.

parcel to cover the horizontal distanceL at the speedU , that is,T = L/U . To climb a heightof ∆z, the fluid needs to acquire a vertical velocity on the order of

W =∆z

T=

U∆z

L. (11.13)

The vertical displacement is on the order of the height of theobstacle and, in the presence ofstratificationρ(z), causes a density perturbation on the order of

∆ρ =

∣

∣

∣

∣

dρ

dz

∣

∣

∣

∣

∆z

=ρ0N

2

g∆z, (11.14)

whereρ(z) is the fluid’s vertical density profile upstream. In turn, this density variation givesrise to a pressure disturbance that scales, via the hydrostatic balance, as

∆P = gH∆ρ

= ρ0N2H∆z. (11.15)

By virtue of the balance of forces in the horizontal, the pressure-gradient force must be ac-companied by a change in fluid velocity [u∂u/∂x+ v∂u/∂y ∼ (1/ρ0)∂p/∂x]:

U2

L=

∆P

ρ0L=⇒ U2 = N2H∆z. (11.16)

From this last expression, the ratio of vertical convergence,W/H , to horizontal divergence,U/L, is found to be

11.5. FROUDE NUMBER 329

W/H

U/L=

∆z

H=

U2

N2H2. (11.17)

We immediately note that ifU is less than the productNH ,W/H must be less thanU/L,implying that convergence in the vertical cannot fully meethorizontal divergence. Conse-quently, the fluid is forced to be partially deflected horizontally so that the term∂u/∂x canbe met by−∂v/∂y better than by−∂w/∂z. The stronger the stratification, the smaller isUcompared toNH and, thus,W/H compared toU/L.

From this argument, we conclude that the ratio

Fr =U

NH, (11.18)

called theFroude number, is a measure of the importance of stratification. The rule is: IfFr . 1, stratification effects are important; the smallerFr, the more important theseeffects are.

The analogy with the Rossby number of rotating fluids,

Ro =U

ΩL, (11.19)

whereΩ is the angular rotation rate andL the horizontal scale, is immediate. Both Froudeand Rossby numbers are ratios of the horizontal velocity scale by a product of frequencyand length scale; for stratified fluids, the relevant frequency and length are naturally thestratification frequency and the height scale, whereas in rotating fluids they are, respectively,the rotation rate and the horizontal length scale.

The analogy can be pursued a little further. Just as the Froude number is a measure of thevertical velocity in a stratified fluid [via (11.17)], the Rossby number can be shown to be ameasure of the vertical velocity in a rotating fluid. We saw (Section7.2) that strongly rotatingfluids (Ro nominally zero) allow no convergence of vertical velocity,even in the presence oftopography. This results from the absence4 of horizontal divergence in geostrophic flows. Inreality, the Rossby number cannot be nil, and the flow cannot be purely geostrophic. The non-linear terms, of relative importance measured byRo, yield corrective terms to the geostrophicvelocities of the same relative importance. Thus, the horizontal divergence,∂u/∂x+∂v/∂y,is not zero but is on the order ofRoU/L. Since the divergence is matched by the verticaldivergence,−∂w/∂z, on the order ofW/H , we conclude that

W/H

U/L= Ro, (11.20)

in rotating fluids. Contrasting (11.17) to (11.20), we note that, with regard to vertical veloci-ties, the square of the Froude number is the analogue of the Rossby number.

In continuation of the analogy, it is tempting to seek the stratified analogue of the Taylorcolumn in rotating fluids. Recall that Taylor columns occur in rapidly rotating fluids (Ro =U/ΩL 1). Let us then ask what happens when a fluid is very stratified (Fr = U/NH

1). By virtue of (11.17), the vertical displacements are severely restricted (∆z H), imply-ing that an obstacle causes the fluid at that level to be deflected almost purely horizontally.

4For the sake of the analogy, we rule out here an eventual beta effect.

330 CHAPTER 11. STRATIFICATION

(In the absence of rotation, there is no tendency toward vertical rigidity, and parcels at levelsabove the obstacle can flow straight ahead without much disruption.) If the obstacle occu-pies the entire width of the domain, such a horizontal detouris not allowed, and the fluid atthe level of the obstacle is blocked on both upstream and downstream sides. This horizontalblocking in stratified fluids is the analogue of the vertical Taylor columns in rotating fluids.Further analogies between homogeneous rotating fluids and stratified nonrotating fluids havebeen described by Veronis (1967).

11.6 Combination of rotation and stratification

In the light of the previous remarks, we are now in position toask what happens when, as inactual geophysical fluids, the effects of rotation and stratification are simultaneously present.The preceding analysis remains unchanged, except that we now invoke the geostrophic bal-ance [see (7.4)] in the horizontal momentum equation to obtain the horizontal velocity scale:

ΩU =∆P

ρ0L=⇒ U =

N2H∆z

ΩL. (11.21)

The ratio of the vertical to horizontal convergence then becomes

W/H

U/L=

∆z

H=

ΩLU

N2H2

=Fr2

Ro. (11.22)

This is a particular case of great importance. According to our foregoing scaling analysis,the ratio of vertical convergence to horizontal divergence, (W/H)/(U/L), is given byFr2,Fr2/Ro, or Ro, depending on whether vertical motions are controlled by stratification orrotation or both (Figure11-6). Thus, ifFr2/Ro is less thanRo, stratification restricts verticalmotions more than rotation and is the dominant process. The converse is true ifFr2/Ro isgreater thanRo.

Note thatRo is in the denominator of (11.22), which implies that the influence of rota-tion is to increase the scale for the vertical velocity when stratification is present. However,since vertical divergence cannot exist without horizontalconvergence (W/H . U/L), thefollowing inequality must hold:

Fr2 . Ro, (11.23)

that is,

U

NH.

NH

ΩL. (11.24)

This sets an upper bound for the magnitude of the flow field in a fluid under given rotation(Ω) and of given stratification (N ) in a domain of given dimensions (L, H). If the velocityis imposed externally (e.g., by an upstream condition), the inequality specifies eitherthe

11.6. ROTATION AND STRATIFICATION 331

Ro

Fr

1

1Fr ∼ 1

Ro∼

1

Fr∼Ro

W/HU/L ∼ 1W/H

U/L ∼ Ro

W/HU/L ∼ Fr2

Three-dimensionalflow

W/HU/L ∼

Fr2

Ro

Flow influencedmostly byrotation

Flow influencedby both rotationand stratification

Flow influencedmostly bystratification

Figure 11-6 Recapitulation of the vari-ous scalings of the ratio of vertical con-vergence (divergence),W/H , to hori-zontal divergence (convergence),U/L,as a function of the Rossby number,Ro = U/(ΩL), and Froude number,Fr = U/(NH).

horizontal or the vertical length scales of the possible disturbances. Finally, if the system issuch that all quantities are externally imposed and that they do not meet (11.24), then specialeffects such as Taylor columns or blocking must occur.

Inequality (11.24) brings a new dimensionless numberNH/ΩL, namely, the ratio of theRossby and Froude numbers. For historical reasons and also because it is more convenient insome dimensional analyses, the square of this quantity is usually defined:

Bu =

(

NH

ΩL

)2

=

(

Ro

Fr

)2

. (11.25)

It bears the name ofBurger number, in honor of Alewyn P. Burger (1927–2003), who con-tributed to our understanding of geostrophic scales of motions (Burger, 1958). In practice,the Burger number is a useful measure of stratification in thepresence of rotation.

In typical geophysical fluids, the height scale is much less than the horizontal lengthscale (H L), but there is also a disparity between the two frequenciesΩ andN . While therotation rate of the earth corresponds to a period of 24 h, thestratification frequency generallycorresponds to much shorter periods, on the order of few to tens of minutes in both the oceanand atmosphere. This implies that generallyΩ N and opens the possibility of a Burgernumber on the order of unity.

Stratification and rotation influence the flow field to similardegrees ifFr2/Ro andRoare on the same order. Such is the case when the Froude number equals the Rossby numberand, consequently, the Burger number is unity. The horizontal length scale then assumes a

332 CHAPTER 11. STRATIFICATION

special value:

L =NH

Ω. (11.26)

For the values ofΩ andN just cited and a height scaleH of 100 m in the ocean and 1 km inthe atmosphere, this horizontal length scale is on the orderof 50 km and 500 km in the oceanand atmosphere, respectively. At this length scale, stratification and rotation go hand in hand.Later on (Chapter15), it will be shown that the scale defined above is none other than theso-calledinternal radius of deformation.

Analytical Problems

11-1. Gulf Stream waters are characterized by surface temperatures around 22C. At a depthof 800 m below the Gulf Stream, temperature is only 10C. Using the value 2.1× 10−4

K−1 for the coefficient of thermal expansion, calculate the stratification frequency.What is the horizontal length at which both rotation and stratification play comparableroles? Compare this length scale to the width of the Gulf Stream.

11-2. An atmospheric inversion occurs when the temperature increases with altitude, in con-trast to the normal situation when the temperature decays with height. This correspondsto a very stable stratification and, hence, to a lack of ventilation (smog, etc.). What isthe stratification frequency when the inversion sets in (dT/dz = 0)? TakeT = 290 KandCp = 1005 m2 s−2 K−1).

11-3. A meteorological balloon rises through the lower atmosphere, simultaneously measur-ing temperature and pressure. The reading, transmitted to the ground station wherethe temperature and pressure are, respectively, 17C and 1028 millibars, reveals a gra-dient∆T/∆p of 6C per 100 millibars. Estimate the stratification frequency.If theatmosphere were neutral, what would the reading be?

11-4. Wind blowing from the sea at a speed of 10 m/s encounters Diamond Head, an extinctvolcano on the southeastern coast of O’ahu Island in Hawai’i. This volcano is 232 mtall and 20 km wide. Stable air possesses a stratification frequency on the order of 0.02s−1. How do vertical displacements compare to the height of the volcano? What doesthis imply about the importance of the stratification? Is theCoriolis force important inthis case?

11-5. Redo Problem 11-4 with the same wind speed and stratificationbut with a mountainrange 1000 m high and 500 km wide.

11-6. Vertical soundings of the atmosphere provided the temperature profiles displayed inFigure11-7. Analyze the stability of each profile.

11.6. ROTATION AND STRATIFICATION 333

k

T(a)

k

T(b)

k

T(c)

k

T(d)

Figure 11-7 Various vertical profilesof temperature (solid lines) with thelapse rate (dashed line) correspondingto a particular fluid parcel (dot).

Numerical Exercises

11-1. Usemedprof.m to read average Mediterranean temperature and salinity vertical pro-files and calculateN2 for various levels of vertical resolution (averaging data withincells). What do you conclude? (Hint: Useies80.m for the state equation.)

11-2. Use the diffusion-equation solver of Numerical Exercise 5-4 with a turbulent diffu-sion coefficient that changes from 10−4 m2/s to 10−2 m2/s wheneverN2 is negative.Simulate the evolution of a 50 m high water column with an initially stable verticaltemperature gradient of 0.3C/m subsequently cooled at the surface by a heat loss of100 W/m2. Salinity is unchanged. Study the effect of changes in∆z and∆t.

11-3. Implement the algorithm outlined in Section11.4, to remove any gravitational instabil-ity instantaneously. Keep the turbulent diffusivity constant at 10−4 m2/s and simulatethe same problem as in Numerical Exercise 11-2.

David Brunt1886 – 1965

As a bright young British mathematician, David Brunt began acareer in astronomy, analyzingthe statistics of celestial variables. Then, turning to meteorology during World War I, hebecame fascinated with weather forecasting and started to apply his statistical methods toatmospheric observations in the search for primary periodicities. By 1925, he had concludedthat weather forecasting by extrapolation of cyclical behavior was not possible and turned hisattention to the dynamic approach, which had been initiatedin the late nineteenth century byWilliam Ferrel and given new impetus by Vilhelm Bjerknes in recent years.

In 1926, he delivered a lecture at the Royal Meteorological Society on the vertical oscil-lations of particles in a stratified atmosphere. Lewis F. Richardson then led him to a paperpublished the preceding year by Finnish scientist Vilho Vaisala, in which the same oscillatoryfrequency was derived. This quantity is now jointly known asthe Brunt–Vaisala frequency.

Continuing his efforts to explain observed phenomena by physical processes, Brunt con-tributed significantly to the theories of cyclones and anticyclones and of heat transfer in theatmosphere. His studies culminated in a textbook titledPhysical and Dynamical Meteorology(1934) and confirmed him as a founder of modern meteorology. (Photo credit: LaFayette,London)

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Vilho V aisala1889 – 1969

Altough he obtained his doctorate in mathematics (at the University of Helsinki, Finland),Vilho Vaisala found the subject rather uninspiring and became interested in meteorology.His positions at various Finnish institutes, including theIlmala Meteorological ObservationStation, required of him to develop instruments for atmospheric observations, which he didwith much ingenuity. This eventually led him to establish in1936 a commercial company forthe manufacture of meteorological instrumentation, the Vaisala Company, a company nowwith branches on five continents and sales across the globe. In addition to his inventions andcommercial activities, Vaisala retained an interest inthe physics of the atmosphere, publish-ing over one hundred scientific papers, and mastered nine foreign languages. (Photo credit:Vaisala Archives, Helsinki)

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